I’m going to start out with one very specific example which is incredibly typical of the show, and then I’m going to go into character profiles of Raj (the Indian character) and Howard (the Jewish character) and what exactly is racist about the way they’re written.
Here’s a piece of dialogue from the Season 5 opener. Penny (White woman) got drunk with Raj (Indian man), and they passed out and woke up in his bed. It should be noted that they’ve been friends for the last four years. She thinks they may have slept together, and confides in her friend Bernadette (White woman)
Amy: Do you know the story of Catherine the Great?
Amy: She ruled Russia in the 1700’s and one night when she was feeling particularly randy she used an intricate system of pulleys to have intimate relations with a horse.
Penny: I’m sorry, what does this have to do with me?
Amy: She engaged in inter-species hanky panky and people still call her great. I’m sure your reputation can survive you shagging a little Indian boy.
Raj: As your friend, you might want to know that we didn’t have sex in the conventional sense.Penny: Oh God, did you pull some weird Indian crap on me?
Yes, Penny, that’s right. All Indian people must be perverts because of the Kama Sutra. Good thing you didn’t let him defile your pure White woman body.
Raj: Indian character. The first thing token hot girl Penny ever says to him is “Oh, do you speak English?” There are constantly jokes made about his small penis, and he is CONSTANTLY cut out of most of the substantive parts of the group dynamic, including hyper-intelligence, fandoms, and comics. (Good article that I don’t entirely agree with from the NYT on how American nerd culture is “hyperwhite” if you’re interested) His parents also push an arranged marriage on him, which is about as likely as one of Ross’s brides on Friends asking for a dowry.
Raj is also massively emasculated on the show - he only likes girly drinks, he’s the “feminine one” in what is described as his “ersatz homosexual” bromance relationship with Howard, he likes Sex and the City, Eat Pray Love, and the Notebook, there are FREQUENT jokes about him having a very small penis, and he’s not only incapable of wooing women, he straight up can’t speak to them.
It’s amazing that they manage to fit all that racism into his character, since he’s pretty much marginalized (someone did a show breakdown in 2011 and found that in all of the seasons of Big Bang Theory to date, Raj had only been involved in FIVE episode subplots total).
Howard Wolowitz: Jewish character. He lives with his screaming banshee of a mother who treats him like he’s still a toddler (another character describes him as having an Oedipal complex). He’s extremely lascivious, and goes to desperate lengths to attempt to get (White) women to sleep with him. Like the other characters on Big Bang Theory, he gets turned down a lot - unlike the other characters, he’s frequently described as “disgusting” or some variation thereof. His pickup attempts often tie into his race - while trying to seduce Sheldon’s sister, he offers to kill his rabbi with a pork chop if she’ll sleep with him.
On top of everything previously mentioned, I feel like I should point out the small detail of clothes - through five seasons, the nerd group of four has maintained a pretty steady style. Howard wears awful and ill-fitted 1960’s-inspired clothes, and Raj wears frumpy argyle sweater vests with a ratty windbreaker. The two White nerds, however, somehow manage to maintain a consistent hipstery-geek chic style.
And, in case I haven’t sold the racism argument - there’s also the matter of sexism on The Big Bang Theory, but that’s a blog post for another day.